http://www.wdwinfo.com/wdwinfo/rehab.htm says closed through June 3.
We were at AK on April 23, which was supposed to be the re-opening date, we were happy because we like Flame Tree. But on our way to EE at RD we saw the construction wall, constructions workers walking in and out, we could see through the door that the entire place was dug up.
Note that other nearby food trucks were advertising that they had Flame Tree BBQ menu items, in case you will be craving something.